Slice the flank steak against the grain into very thin, bite-sized strips.
In a small mixing bowl, whisk together the tamari, coconut aminos, and toasted sesame oil to create the stir-fry sauce.
Heat the avocado oil in a large skillet or wok over medium-high heat until shimmering.
Add the beef strips to the hot pan in a single layer and sear without moving for 1 minute, then stir and cook for another minute until browned; remove beef from the pan and set aside.
Add the broccoli florets to the same pan with two tablespoons of water, cover with a lid, and steam for 2 minutes until bright green and tender-crisp.
Remove the lid and stir in the minced garlic, grated fresh ginger, and red pepper flakes, sautéing for 30 seconds until fragrant.
Return the cooked beef and any accumulated juices back into the pan.
Pour the prepared sauce over the beef and broccoli, tossing constantly for 1 minute until the sauce thickens slightly and coats everything evenly.
Place the warm cooked brown rice in a bowl and top with the beef and broccoli mixture.
Garnish with toasted sesame seeds before serving.